Oxford Mathematician Patrick Kidger writes about combining the mathematics of differential equations with the machine learning of neural networks to produce cutting-edge models for time series.
How to deal with resistance? This is the headline question these days with regards to COVID vaccines. But it is an important question also in cancer therapy. Over the past century, oncology has come a long way, but all too often cancers still recur due to the emergence of drug-resistant tumour cells. How to tackle these cells is one of the key questions in cancer research. The main strategy so far has been the development of new drugs to which the resistant cells are still sensitive.
As we have seen over the past year, the ability to accurately predict the course of an epidemic is imperative when making decisions about how to control the spread of a virus.
Take a piece of rope and knot it as you wish. When you are done, glue the two extremities together and you will obtain a physical realisation of what mathematicians also call a knot: a simple closed curve in 3-dimensional space. Now, put the knotted rope on a table and take a picture of it from above. It is now a planar projection of your knot. The mathematical equivalent of it is a knot diagram with multiple crossings as shown in the figure.
Oxford Mathematician Vladimir Markovic talks about his research into intrinsic geometry of Teichmüller Spaces.
Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.
Oxford Mathematician Markus Dablander talks about his collaboration with Julius Berner and Philipp Grohs from the University of Vienna. Together they developed a new deep-learning-based method for the computationally efficient solution of high-dimensional parametric Kolmogorov PDEs.
Oxford Mathematician Daniel Woodhouse talks about the theorem that motivates much of his research.
Oxford Mathematician Dawid Kielak talks about his joint work with Marek Kaluba and Piotr Nowak in which they used a computer* to prove a theorem about the behaviour of other computers.
Tissue oxygenation plays a crucial role in the growth of cancerous tumours and their response to treatments. While it may seem intuitive that reducing oxygen delivery to a tumour would be a treatment therapy, low oxygen levels (hypoxia) can significantly reduce the effectiveness of treatments such as radiotherapy and some chemotherapies. Therefore, understanding the dynamics of a tumour's red blood cells - which carry oxygen through the vasculature - is of vital importance.
Oxford Mathematician Kristian Kiradjiev talks about his DPhil research, supervised by Chris Breward and Ian Griffiths in collaboration with W. L. Gore and Associates, Inc., on modelling filtration devices for removal of sulphur dioxide from flue gas.
When Oxford Mathematician Alain Goriely was approached by his collaborator Ellen Kuhl from Stanford University to work on a travel restriction issue in Newfoundland he started a Coronavirus journey that ended up in the Canadian Supreme Court.
Oxford Mathematician Daniel Gulotta talks about his work on $p$-adic geometry and the Langlands program.
"Geometry is one of the more visceral areas of mathematics. Concepts like distance and curvature are things that we can actually see and feel.
Oxford Mathematician Artur Ekert describes how his research in to using Quantum properties for cryptography led to some very strange results.
Oxford Mathematician Vidit Nanda discusses his recent work with colleagues Bernadette Stolz, Jared Tanner and Heather Harrington on detecting singularities in data.
One of the great puzzles of the current COVID-19 crisis is the observation that older people have a much higher risk of becoming seriously ill. While it is usually commonly accepted that the immune system fails progressively with age, the actual mechanism leading to this effect was not fully understood. In a recent work, Sam Palmer from Oxford Mathematics and his colleagues in Cambridge have proposed a simple and elegant solution to this puzzle.
Mathematical models have been used throughout the COVID-19 pandemic to help plan public health measures. Attention is now turning to how interventions can be removed while continuing to restrict transmission. Predicting the effects of different possible COVID-19 exit strategies is an important current challenge requiring mathematical modelling, but many uncertainties remain.
A set of integers greater than 1 is primitive if no number in the set divides another. Erdős proved in 1935 that the series of $1/(n \log n)$ for $n$ running over a primitive set A is universally bounded over all choices of A. In 1988 he conjectured that the universal bound is attained for the set of prime numbers. In this research case study, Oxford's Jared Duker Lichtman describes recent progress towards this problem: